To identify which among the given equations is a quadratic inequality, we need to understand the nature of each equation.
1. [tex]\( x^2 + 5x + 10 > 25 \)[/tex]
- This equation involves the term [tex]\( x^2 \)[/tex], which signifies that it is a quadratic equation.
- Additionally, the inequality symbol [tex]\( > \)[/tex] makes it a quadratic inequality.
2. [tex]\( 3x + 7 > 12 \)[/tex]
- This equation is linear because the highest power of [tex]\( x \)[/tex] is 1.
- Therefore, it is a linear inequality.
3. [tex]\( 5x^2 + 3x + 12 = 0 \)[/tex]
- This equation involves the term [tex]\( x^2 \)[/tex], making it a quadratic equation.
- However, it uses an equality symbol [tex]\( = \)[/tex], not an inequality symbol, so it is not an inequality.
4. [tex]\( x^3 + 12 < -3 \)[/tex]
- This equation involves the term [tex]\( x^3 \)[/tex], making it a cubic equation.
- It uses an inequality symbol [tex]\( < \)[/tex], so it is a cubic inequality.
From the above analysis, we can identify that the first equation:
[tex]\[ x^2 + 5x + 10 > 25 \][/tex]
is the quadratic inequality.
In conclusion, the quadratic inequality among the given equations is the first one, located in position 1.
